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# Expectation and Variance

The expectation of a random variable,
$E(X)$
, is the average of possible values weighted by their probabilities. Formally, it can be defined in two ways:
1. 1.
Domain definition:
$E(X) = \sum_{\omega \in \Omega} X(\omega) P(\omega)$
.
2. 2.
Range definition:
$E(X) = \sum_x x P(X = x)$
.
Expectation has nice properties of linearity:
$E(X + Y) = E(X) + E(Y)$
and
$E(aX + b) = aE(x) + b$
.